How to capture lighting in video shots?
While there are many subtleties and nuances to your question, the following should deliver reasonable results with equipment you may very well have around you or possibly borrow.

Use a DSLR and use your widest lens. The Sigma 8mm fisheye and a full 35mm sensor allows one to capture edge to edge sphere shots, for example.
Do a bracketed exposure series of your set in enough directions to cover the scene. For example, with the 8mm above, it is three directions. If you require four photos to deliver your 360 degree view, you would require a set for each exposure. Depending on your scene's dynamic range, pan to your first position and take photos at exposure, several stops down, and several stops up. -3, -1.5, 0, +1.5, +3 stops might be required, or fewer. Shoot raw to preserve as great a range as possible for each image.
Merge each bracketed exposure set together into a spherical merged image. An example might be Hugin or other such tool.
Merge the stacks of exposures into a single HDR scene referred EXR image from the layers of stitches. An example might be Luminance HDR or other such tool.

A single exposure will fail to deliver upon the &quot;correctly lit&quot; aspect of your question.
It is important to understand the difference between a scene referred and a display referred image here. Where a typical JPEG or such image is a low dynamic range and being bent to fit the 0..1 range, a scene referred image attempts to maintain the original scene's ratios of light levels being cast from objects. This is stored in a linear fashion and the values can range from zero to a theoretical infinity.
When a series of bracketed photos are merged into a scene referred file format such as EXR, the original values of the scene are reconstructed. These values will much wider apart than the 2.5 stops of light a typical JPG or PNG can represent1.
Granted that the above will only deliver an accurate estimation of the lighting at the given point you photographed your HDRI images. If you seek more accuracy such as a CG element nears a wall or other object, you would need to attempt to project the light values upon a placebo object to correctly estimate the inverse square law of energy (1/d^2). Project from view with some tracking might be useful here.
For further reading:

The VES technology white paper discussing the differences between scene referred and display referred data.
A solid description of the differences of file formats and their ability to represent HDR data.
Hugin panoramic and spherical stitching application.
Luminance HDR: http://qtpfsgui.sourceforge.net/

1: A middle grey card photographed and converted to a JPEG will land at approximately 48.45% value in sRGB. This translates to a value of 0.2 when linearized to display linear. If we double this, we arrive at 0.4 for one stop. Doubled again to 0.8 for two stops. 1.0 therefore can't express any greater dynamic range than a little over two and a tenth stops of light. Further complicating this is the fact that a standard camera will hold a wider dynamic range and curve this into the resulting JPEG or display referred format. This ends up baking both the sRGB curve and the camera curve into the resulting image, and it cannot be reasonably undone from these formats.